Intersection theory on moduli of smooth complete intersections

We provide a general method for computing rational Chow rings of moduli of smooth com-plete intersections. We specialize this result in different ways: to compute the integral Picard group of the associated stack; to obtain an explicit presentation of rational Chow rings of moduli of smooth complete intersections of codimension two; to prove old and new results on moduli of smooth curves of genus = 5 and polarized K3 surfaces of degree = 8.